I am planning a new home network setup and would like your feedback. My current idea is to use my TP-Link Archer AXE75 configured as an Access Point for WiFi coverage. For DNS, I will run AdGuard Home on a Banana Pi BPI-R4 Pro with OpenWRT, which will act as my primary router and primary DNS server. As a fallback, I plan to keep my Raspberry Pi 5 (8 GB) as a backup DNS server.
Do you think this is a solid setup, or are there any improvements you would recommend?
Not very solid, but if you have all the components in hand, better use would be to run eg debian with containers for agh and similar on raspberry while keep banana only for routing (and wifi I assume?)
Probably broadcom-based AXE will work until tplink decides to upgrade and remove needed bridge without dhcp functions.
It's not a particularly sensible hardware selection, nor economically feasible (not even technically, unless you really know what you're doing) - on the contrary. I'd go back to the drawing board (as neither of the mentioned devices is really great for the intended purpose).
You implied kind of that you have the 3 devices?
If you have raspberry -> run "applications" in containers
If you have banana -> be it router
If you have axe -> try to scam seller into DoA
What is the total area you are trying to cover?
Longest diagonal distance?
Average wall material?
It's not clear what you already own/ use (apart from the axe75, which is 'not great') and what you only have in mind to buy - nor really what you expect those devices to do.
If you need wireless, the BPi-r4 would not be my choice (at all), if you're only looking for a wired router, x86_64 (alderlake-n, n100, n97, n150, …) would rate much higher on my list, while being in the same price range (slightly different preferences for 10 GBit/s ethernet).
The RPi5 is overpriced for this use case - and I really don't get what you mean by "backup DNS server". If you go all enterprise/ HA, sure you want backup DNS as well, but that's a rather complex setup (and will not work with disjunct implementations, as both need to talk to each other - and understand each other in order not to create a real mess) - if you have to ask, that's not what you should be looking at. Furthermore, what are the chances that your DNS breaks without the router underneath? It's possible, yes - but (even for ambitious-) home use you're optimizing the wrong end and effectively destabilize your system more than you'd harden it.
The AXE75 is Broadcom based, so it will never run OpenWrt - at the same time I don't think it supports any of the fun stuff (multiple VLANs, multi-ESSID, which will become a topic much earlier than a backup DHCP/ DNS server).
KISS is a really important factor in home networking, a lot of things are possible, but they aren't always sensible. But we have far too little information about your requirements to provide effective guidance, just that the prospected hardware is 'always' overpriced and 'not-ideal'.